Download after Nash eqm, Subgame Perfect Nash eqm, and Bayesi

Our last equilibrium concept
The last equilibrium concept we’ll study — after Nash eqm, Subgame Perfect
Nash eqm, and Bayesian Nash eqm — is Perfect Bayesian Equilibrium.
• Perfect refers to the fact that the game will be dynamic, like the kind we
solved using Subgame Perfect Nash Equilibrium
• Bayesian refers to the fact that the game will include incomplete information, so that players are unsure of other players’ private information or
past decisions.
There are actually other important solution concepts — Sequential Equilibrium, Markov Perfect Equilibrium — that won’t be covered in the course, but
are commonly used. PBE was developed in the 1980’s, and is a pretty “cutting
edge” idea, given that Nash Equilibrium was discovered in 1950 by Nash, Subgame Perfect Nash Equilibrium was discovered in 1965 by Reinhard Selten, and
Bayesian Nash Equilibrium was discovered in 1967 by John Harsanyi.
Information Sets in the Extensive Form
Definition 1. An information set is a set of decision nodes, all belonging to
the same player, over which that player cannot distinguish.
Consider this version of the Battle of the Sexes game:
• The woman W decides whether to Go Out on a date or Stay in.
• If W goes out, she goes to a football game F or the ballet B.
• If W goes out, she leaves a message for M saying that she’s going to meet
him, but it doesn’t say where, and her phone goes straight to voice mail.
The man M can go to F or B.
How can we represent this game in an extensive form?
Example
The information set is represented by the dashed line.
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Subgames
Definition 2. A subgame is all of the nodes, branches, and payoffs that follow
after a singleton information set.
So in this game:
there are TWO subgames, not three. We cannot “cut” information sets up.
Beliefs
Definition 3. A set of beliefs are a probability distribution over all the nodes
in a given information set.
Note that “beliefs” is a technical term, like the word “heat” or “energy”
in physics. We don’t mean that beliefs are “feelings” (feelings are not real in
economics) or anything like that. The word just suggests the right idea about
the probability distribution being a subjective assessment of likelihoods.
Beliefs
For the example, we have µ and 1 − µ as the beliefs:
So µ is (the probability that W chose F , given that she went out).
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Perfect Bayesian Equilibrium
∗
Definition 4. A perfect Bayesian equilibrium is a set of strategies σ ∗ = (σ1∗ , ..., σN
)
and beliefs for every player at every information set, so that
• Bayesian Beliefs: The beliefs are derived from the strategies and common
prior beliefs using Bayes’ rule, ¡wherever possible!
• Sequential Rationality: The strategies σ ∗ are optimal at every point in the
game, given the players’ beliefs.
Note that we use the strategies to derive the beliefs, but the beliefs must be
consistent with the strategies.
Perfect Bayesian Equilibrium
Then for
a Perfect Bayesian Equilibrium is a set of strategies for M and W , and beliefs
µ∗ for M , so that no player has a profitable deviation.
Equilibrium 1
Let’s think about this strategy profile:
• W goes out.
• W goes to B.
• M goes to B.
• µ∗ = 0
Is this a perfect Bayesian equilibrium if x = 0? x = 1/2? x = 3/2?
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Equilibrium 2
Let’s think about this strategy profile:
• W goes out.
• W goes to F .
• M goes to F .
• µ∗ = 1
Is this a perfect Bayesian equilibrium if x = 0? x = 1/2? x = 3/2?
Equilibrium 3
Let’s think about this strategy profile:
• W goes out.
• W goes to F with probability 1/3, and B with probability 2/3.
• M goes to F with probability 2/3, and B with probability 1/3.
• µ∗ = 1/3
Is this a perfect Bayesian equilibrium if x = 0? x = 1/2? x = 3/2?
Example
Let’s find some PBE’s from scratch:
Signaling Games
The most interesting class of games that are solved used the perfect Bayesian
Equilibrium concept are signaling games:
• (i) Nature assigns a privately known type to the Sender
• (ii) The sender chooses a message to send to the Receiver
• (iii) The receiver takes an action that determines both their payoffs
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Signaling Games
The “fun part” of signaling games is that the message often has nothing to
do with the information the sender has per se, but the signals simply vary in
cost:
• Educational signaling: (i) a student finds out whether he low or high
productivity, (ii) the student chooses to get a high school or college degree,
and (iii) a firm decides what wage to offer.
• Lawsuits: (i) a client finds out whether he has a genuine or spurious
lawsuit against a firm, (ii) the client chooses to get an expensive or cheap
lawyer, and (iii) the firm decides whether to settle or go to court.
• Advertising and Introductory Offers: (i) a firm learns whether its new
product is good or bad, (ii) the firm chooses to advertise/discount the
product or not, and (iii) consumers decide whether to try it out or not.
• Limit pricing: (i) a firm finds out whether he is low- or high-cost, (ii) the
firm chooses whether to charge a low or a high price, and (iii) a potential
entrant decides whether to enter the market or not.
Separating, Pooling, and Hybrid Equilibria
There are two “kinds” of perfect Bayesian equilibria we might find in signaling games:
• Separating equilibria: All sender types choose different messages to send
to the receiver.
• Pooling equilibria: Multiple sender types send the same message to the
receiver.
• Hybrid equilibria: Some types mix over the signal they send, while others
use pure strategies. (We won’t cover these)
Separating equilibria are “well-behaved”, while pooling equilibria can be more
complicated to analyze.
Two Examples: Timing
• Nature gives the sender type S1 with probability p and type S2 with
probability 1 − p
• The sender sends message u or message d
• Knowing the message — but not the sender’s true type — the receiver
chooses action ℓ or action r
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Example 1
Here’s a signaling game that has two separating equilibria:
Eqm 1: Our first separating equilibrium has the strategies:
• Type S1 chooses u, type S2 chooses d
• R plays ℓ if he sees u, and r if he sees d
Or
5, 2
1, 0
ℓ
r
p
N
d[b]
ℓ
2, 3
1−p
r
ℓ
−1, 0
where the receiver’s beliefs are given by:
a = pr[S1 |u]
1 − a = pr[S2 |u]
b = pr[S1 |d]
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[1 − a]
u
S2
[1 −db]
R
1, 2
r
ℓ
R
[a]
u
S1
5, 2
0, 0
r
3, 3
1 − b = pr[S2 |d]
Now, given these strategies, we derive the beliefs using Bayes’ rule:
a = pr[S1 |u] =
pr[S1 ∩ u]
p
= =1
pr[u]
p
pr[S1 ∩ d]
0
=
=0
pr[d]
1−p
These beliefs are Bayesian, so an equilibrium with a∗ = 1 and b∗ = 0 will satisfy
that part of the criteria. Let’s check sequential rationality:
b = pr[S1 |d] =
• Type S1 has no profitable deviation, because if he sends the message d rather
than u, the receiver then believes he is facing an S2 type and plays r, and S1
then gets a payoff of 1 rather than 5.
• Type S2 has no profitable deviation, because if he sends the message u rather
than d, the receiver then believes he is facing an S1 type and plays ℓ, and S2
then gets a payoff of 0 rather than 3.
• At the bottom information set, given R’s beliefs, R get a payoff of 3 from
r and 0 from ℓ, so the receiver has no profitable deviation at the bottom
information set. At the top information set, given R’s beliefs, R gets a payoff
of 2 from ℓ and 0 from r, so the receiver has no profitable deviation at the
top information set.
That covers all the cases. Then S1 plays u, S2 plays d, R plays r if d is played and
R plays ℓ if u is played, a∗ = 1, b∗ = 0 is a perfect Bayesian equilibrium of the
game. It happens to be a separating equilibrium.
See if you can figure out the other separating equilibrium. Ask me if you have
questions.
Example 2
Here’s a signaling game that potentially has two pooling equilibria:
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Consider the strategies: Both S1 and S2 send message d; R chooses r if d is
played, and ℓ if u is played.
1, 1
3, 0
ℓ
r
R
[a]
u
S1
d[b]
ℓ
p
N
1−p
ℓ
2, 0
r
ℓ
R
r
0, 1
2, 1
0, 0
1, 0
[1 − a]
u
S2
[1 −db]
r
3, 1
Then the receiver’s beliefs should be
b = pr[S1 |d] =
pr[S1 ∩ d]
p
=
=p
pr[d]
p + (1 − p)
pr[S1 ∩ u]
0
= =?
pr[u]
0
So the ¡wherever possible! scenario has finally shown up to the party. Asking,
“What’s the probability that S1 sends u?” here is like asking, “What’s the probability a good job market candidate doesn’t wear a suit to an interview?” It never
happens, but we can still pose the question. So we will solve for the range of beliefs
that make sense of the situation, since Bayes’ rule is no help.
Let’s check sequential rationality (and solve for a):
a = pr[S1 |u] =
• The S1 type has no incentive to deviate, since if he sends u instead of d, the
receiver plays ℓ, and the S1 type gets a payoff of 1 rather than 2, which is
strictly worse. So S1 has no profitable deviation.
• The S2 type has no incentive to deviate, since if he sends u instead of d, the
receiver plays ℓ, and the S2 type gets a payoff of 0 rather than 3, which is
strictly worse. So S2 has no profitable deviation.
• At the bottom information set, for this to be an equilibrium, the expected
payoff to r must be greater than the expected payoff to ℓ, or
p(0) + (1 − p)1 ≥ p(1) + (1 − p)(0)
implying that p ≤ 1/2. At the top information set, for this to be an equilibrium, the expected payoff to ℓ must be greater than the expected payoff to
r, or
a(1) + (1 − a)(0) ≥ a(0) + (1 − a)(1)
implying that a ≥ 1/2.
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Then the strategies S1 and S2 send message d; the receiver responds to d with
r and to u with ℓ; and the receiver’s beliefs are b∗ = 0 and a ≥ 1/2; are a perfect
Bayesian equilibrium if p ≤ 1/2. (simple, right?)
See if you can find another pooling pBe in this game. Ask if you have questions.
Example: Educational Signaling
• Nature assigns a productivity type — High or Low — to the worker.
• The worker chooses to go to High School only, HS, or High School and
College, C.
• The firm observes the worker’s education, but not the worker’s type, and
offers either a high wage, wh , or a low wage, wl .
Educational Signaling: Extensive Form
Educational Signaling: Separating Equilibria (?)
Is there a separating equilibrium of the following type:
• High types send C, low types send HS.
• The firm pays the high wage to applicants with degree C, and the low
wage to applicants with degree HS.
• In equilibrium, the firm believes only high types go to college, and only
low types go to high school.
Let’s put beliefs in the extensive form:
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2, 2
1, 1
wh
wl
Firm
[a]
HS
High
p
C[b]
N
Firm
2, 0
1, 1
wh
wl
[1 − a]
HS
1−p
Low
C
[1 − b]
wh
wl
wh
wl
3, 2
1, 1
2, −1
0, 1
Then the beliefs are
a = pr[High|HS]
1 − a = pr[Low|HS]
b = pr[High|C]
1 − b = pr[Low|C]
OK... what do we need to do? Well, we have the proposed strategies: High types
go to college, low types go to high school; the firm pays the high wage to college
grads, the low wage to high school grads. If these strategies are actually adopted
in equilibrium, we have
2, 2
1, 1
wh
wl
Firm
[a]
HS
High
p
C[b]
N
2, 0
1, 1
wh
wl
1−p
Firm
[1 − a]
HS
Low
[1 −Cb]
wh
wl
wh
wl
3, 2
1, 1
2, −1
0, 1
Given these strategies, we can almost read the beliefs off the above extensive
form:
0
pr[High ∩ HS]
=
=0
a = pr[High|HS] =
pr[HS]
1−p
1 − a = pr[Low|HS] =
pr[Low ∩ HS]
1−p
=
=1
pr[HS]
1−p
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pr[High ∩ C]
p
= =1
pr[C]
p
b = pr[High|C] = pr[High|C] =
1 − b = pr[Low|C] =
pr[Low ∩ C]
0
= =0
pr[C]
p
Nice. OK, so we have strategies and beliefs. Can any type profitably deviate, given
the strategies and beliefs?
• The firm gets an expected payoff of 3 by paying wh to college grads; switching
to wl gives a payoff of 1, which is strictly lower. The firm gets an expected
payoff of 1 by paying wl to high school grads; switching to wh gives a payoff
of 0, which is strictly lower. So given the firm’s beliefs, its strategy is optimal.
• The high type gets a payoff of 3 from going to college. By switching to high
school, it gets a payoff of 1 (since the firm infers that it must be a low type,
and pays wl ). This is an unprofitable deviation, so the high types don’t want
to deviate.
• The low type gets a payoff of 1 from going to high school. By switching to
college, it gets a payoff of 2 (since the firm infers that it must be a high type,
and pays wh ). This is a profitable deviation, so the low types will never go
along with the proposed equilibrium.
Therefore, this game does not have a separating equilibrium in which the degree
is fully informative.
Educational Signaling: Pooling Equilibria
Is there a separating equilibrium of the following type:
• High types send C, low types send C.
• The firm pays the high wage to everyone.
• In equilibrium, the firm has beliefs that are correct.
Again, let’s put beliefs in the extensive form:
2, 2
1, 1
wh
wl
Firm
[a]
HS
High
p
C[b]
N
Firm
2, 0
1, 1
wh
wl
[1 − a]
HS
1−p
Low
C
[1 − b]
wh
wl
wh
wl
3, 2
1, 1
2, −1
0, 1
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Then the beliefs are
a = pr[High|HS]
1 − a = pr[Low|HS]
b = pr[High|C]
1 − b = pr[Low|C]
Now we have the proposed strategies: High types go to college, low types go to
college; the firm pays the high wage to college grads, the low wage to high school
grads. If these strategies are actually adopted in equilibrium, we have
2, 2
1, 1
wh
wl
Firm
[a]
HS
High
p
1, 1
wh
wl
1−p
N
C[b]
2, 0
Firm
[1 − a]
HS
Low
C
[1 − b]
wh
wl
wh
wl
3, 2
1, 1
2, −1
0, 1
Given these strategies, we use Bayes’ rule to get beliefs:
a = pr[High|HS] =
pr[High ∩ HS]
0
= =?
pr[HS]
0
1 − a = pr[Low|HS] =
b = pr[High|C] = pr[High|C] =
1 − b = pr[Low|C] =
pr[Low ∩ HS]
0
= =?
pr[HS]
0
p
pr[High ∩ C]
=
=p
pr[C]
p+1−p
pr[Low ∩ C]
1−p
=
=1−p
pr[C]
p+1−p
But yikes, 0/0? Bayes’ rule doesn’t work on events that don’t happen in equilibrium:
No one sends the HS message, so the probability of receiving the HS message is
zero, and that aspect of beliefs is not well-defined in equilibrium.
Educational Signaling: Beliefs “on the equilibrium path”
To check the firm’s beliefs, we need to compute
pr[High|C] =
p(1)
pr[High ∩ C]
=
=p
pr[C]
p(1) + (1 − p)1
which uses Bayes’ rule, and uses the reasoning from the lecture on dartboards
and broken machine parts.
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Educational Signaling: Beliefs “off the equilibrium path”
Now, we need to compute the firm’s beliefs for every information set, including the one that is never reached in equilibrium. In particular,
pr[High|HS] =
pr[High ∩ HS]
0
= = ...?
pr[HS]
0
This is why the “Bayesian Beliefs” part of the definition of a PBE said ¡wherever
possible!
Educational Signaling: Beliefs “off the equilibrium path”
Off the equilibrium path, we solve for the entire set of beliefs that are consistent with the equilibrium strategies. In particular, we just need the firm to
believe that offering wl in response to HS is better than offering wh :
pr[High|HS]1 + pr[Low|HS]1 ≥ pr[High|HS]2 + pr[Low|HS]0
or
1
≥ pr[High|HS]
2
So facing the game
2, 2
1, 1
wh
wl
Firm
[a =?]
HS
High
p
N
C[p]
Firm
2, 0
1, 1
wh
wl
[1 − a =?]
HS
1−p
Low
[1 −Cp]
wh
wl
wh
wl
3, 2
1, 1
2, −1
0, 1
We ask (i) on the equilibium path, does any type have a profitable deviation, and
(ii) off the equilibrium path, what set of beliefs make the proposed equilibrium satisfy
sequential rationality ? So, since Bayes’ rule doesn’t give us well-defined beliefs, we
then ask, “well, are there any beliefs that work with the proposed strategies? What
do they look like? Are they reasonable or not?”
• The firm gets an expected payoff of p2 + (1 − p)1 by paying wh to college
grads, and p(1) + (1 − p)(1) = 1 by paying wl to college grads. As long as
paying wh is better than paying wl , the firm has no reason to deviate from
the proposed equilibrium, so we need
p2 + (1 − p)1 ≥ 1
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but this is equivalent to
1+p≥1
So the firm will never want to deviate from paying wh to college grads.
• For the firm to want to pay the low wage to high school grads rather than the
high wage, we need the expected payoff of wl to be greater than the expected
payoff of wh , or
a(1) + (1 − a)(1) ≥ a(2) + (1 − a)0
or
pr[High|HS] = a ≤
1
2
• Now, do the sender types have any incentive to deviate? If the high type
sends HS instead of C, it gets a payoff of 1 instead of 2; the high type has
no incentive to deviate. If the low type sends HS instead of C, it gets a
payoff of 1 instead of 2; the low type has no incentive to deviate.
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Therefore, if pr[High|HS] ≤ , there is a (pooling) perfect Bayesian equilibrium
2
where the high and low types both go to college, the firm always pays the high wage,
and pr[High|C] = p.
Educational Signaling: Pooling Equilibria
The follow strategies and beliefs are a perfect Bayesian equilibrium of the
signaling game:
• All sender types choose C.
• The receiver pays wh for C and wl for HS.
• The receiver’s beliefs are pr[High|C] = p, pr[Low|C] = 1 − p, and
pr[High|HS]
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1
≥
2